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Euler's method is a method of numerically estimating a first-order differential equation, given an initial value. It is useful in cases in which a differential equation would be difficult or impossible to solve.

Euler's method calculates an approximate curve by using the slope of the tangent line at any given point. For example, suppose we have the problem

$\frac{dy}{dx} = y, \quad y(0) = 1$

We can break the curve into small line segments. The smaller the segments, the more closely the curve will be approximated. In this case, will use segments of $\Delta x = 1$.

$y(0) = 1$
$y(1) = y(0) + \frac{dy}{dx}(\Delta x) = 1 + y(0) (1) = 2$
$y(2) = y(1) + \frac{dy}{dx}(\Delta x) = 2 + y(1) (1) = 4$

The following table shows the first five values for the function and how they compare to the values of the actual function, ex.

 x y y' ex 0 1 1 1 1 2 2 2.718 2 4 4 7.389 3 8 8 20.086 4 16 16 54.598 5 32 32 148.413
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